Properties

Label 4-325e2-1.1-c3e2-0-1
Degree $4$
Conductor $105625$
Sign $1$
Analytic cond. $367.704$
Root an. cond. $4.37899$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 11·4-s − 30·7-s − 36·8-s + 53·9-s + 78·13-s − 180·14-s − 267·16-s + 318·18-s + 468·26-s − 330·28-s + 288·29-s − 558·32-s + 583·36-s − 606·37-s − 222·47-s − 11·49-s + 858·52-s + 1.08e3·56-s + 1.72e3·58-s + 752·61-s − 1.59e3·63-s + 895·64-s − 72·67-s − 1.90e3·72-s − 2.19e3·73-s − 3.63e3·74-s + ⋯
L(s)  = 1  + 2.12·2-s + 11/8·4-s − 1.61·7-s − 1.59·8-s + 1.96·9-s + 1.66·13-s − 3.43·14-s − 4.17·16-s + 4.16·18-s + 3.53·26-s − 2.22·28-s + 1.84·29-s − 3.08·32-s + 2.69·36-s − 2.69·37-s − 0.688·47-s − 0.0320·49-s + 2.28·52-s + 2.57·56-s + 3.91·58-s + 1.57·61-s − 3.17·63-s + 1.74·64-s − 0.131·67-s − 3.12·72-s − 3.52·73-s − 5.71·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(105625\)    =    \(5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(367.704\)
Root analytic conductor: \(4.37899\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 105625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.808167926\)
\(L(\frac12)\) \(\approx\) \(4.808167926\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad5 \( 1 \)
13$C_2$ \( 1 - 6 p T + p^{3} T^{2} \)
good2$C_2$ \( ( 1 - 3 T + p^{3} T^{2} )^{2} \)
3$C_2^2$ \( 1 - 53 T^{2} + p^{6} T^{4} \)
7$C_2$ \( ( 1 + 15 T + p^{3} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 358 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 7801 T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 13682 T^{2} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 1910 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 144 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 10114 T^{2} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 303 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 100978 T^{2} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 149605 T^{2} + p^{6} T^{4} \)
47$C_2$ \( ( 1 + 111 T + p^{3} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 126358 T^{2} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 138274 T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 - 376 T + p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 36 T + p^{3} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 588373 T^{2} + p^{6} T^{4} \)
73$C_2$ \( ( 1 + 1098 T + p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 830 T + p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 - 438 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 1218094 T^{2} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 852 T + p^{3} T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.87106442819228722157706335898, −11.07995434205870930926470334777, −10.41203109147594006694007539688, −10.11178553201213220270499347407, −9.620028170308798122243178855427, −9.129556173315592289881134145777, −8.662179121947815531624380167550, −8.183092059244316681605186760341, −7.11962844120745145542551012033, −6.62160625949662040169102810198, −6.52403135380960089224769454620, −5.98496614713293216223272105786, −5.28320919903726654455402996808, −4.80221803568983395112351523134, −4.22622582134335857619391331233, −3.75254845520954672112358233845, −3.38951864799546477142079900616, −2.91069101221838435288395217451, −1.68413095921178438608396351490, −0.57254729840677549771603016597, 0.57254729840677549771603016597, 1.68413095921178438608396351490, 2.91069101221838435288395217451, 3.38951864799546477142079900616, 3.75254845520954672112358233845, 4.22622582134335857619391331233, 4.80221803568983395112351523134, 5.28320919903726654455402996808, 5.98496614713293216223272105786, 6.52403135380960089224769454620, 6.62160625949662040169102810198, 7.11962844120745145542551012033, 8.183092059244316681605186760341, 8.662179121947815531624380167550, 9.129556173315592289881134145777, 9.620028170308798122243178855427, 10.11178553201213220270499347407, 10.41203109147594006694007539688, 11.07995434205870930926470334777, 11.87106442819228722157706335898

Graph of the $Z$-function along the critical line